Theorem imaeq2 4885

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4822	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4776	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4560	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4560	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2198	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1332   ran crn 4548   |` cres 4549   "cima 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by:  imaeq2i  4887  imaeq2d  4889  ssimaex  5490  ssimaexg  5491  isoselem  5729  f1opw2  5984  fopwdom  6738  ssenen  6753  fiintim  6825  fidcenumlemrk  6850  fidcenumlemr  6851  sbthlem2  6854  isbth  6863  ennnfonelemp1  11955  ennnfonelemnn0  11971  ctinfomlemom  11976  ctinfom  11977  tgcn  12416  iscnp4  12426  cnpnei  12427  cnima  12428  cnconst2  12441  cnrest2  12444  cnptoprest  12447  txcnp  12479  txcnmpt  12481  metcnp3  12719